214 PEOCEEDINGS OF THE AMERICAN ACADEMY 



for ?• = 0, 1, 2, fi, .. ..V. Since /"g' (x) =/"!,"' (x), either of these 



two functions may be denoted simply by/o (x)- We then have 



/o W +/?' (^) +/r' (^) + • • • -/l" (^) +/f W = 1 ; 



for jt? = 1, 2, and r = 0, 1, 2, . . ,. /x, . . ..v ; 



ff (0) • /!'' (^) = 0, 



for p, q = 1,2, and r, s = 0, 1, 2, jn, v, but r i^: s ; and 



tr./o (^) = /o (tr. 0) 

 = /o(-^) 



tr.r(^)-/:.^tr.^) 



for r = 1 , 2, 3 /x, v. We also have 



^ = ./, (6) + /h V=1/?' ^ - /^i V^/f ^ + . . . . 



therefore, if y(^) denotes any polynomial in powers of or conver- 

 gent power series in 0, 



+ ....+ f(h. V=i)/l^' (0) +/(- /^ V~i)f:' {0). 



Thus 



From the relations given above between the functions with the same 

 subscript it is evident that this matrix is orthogonal. 

 Let now 



